Q(x)*Q(y)=Q(x*y)
but here the input the root of 5 instead of sum variable

??

What is a field, you probbably already know. However, In order to show that it is a ring, alll we need to show is that it is closed under subtraction, and under multiplication. Like Halls already told you.

What you are doing there, is basically a hommomorphism, that is the perservation of the operation, but we are not dealing with mapps between two fields in here.

So let x, and y be any two elements on that set, then obviously they will have the form

i dont know how to construct these equations because there is no variable input?

you don't need to construct these, because you are not dealing with a mapping between two rings or fields, and thus you are not asked to show whether that kind of mapping is a homomorphism. because, those expressions are used when we want to show that a particular mapping say between two rings is a homomorphism.

ring is like x=(a+b,c,g+1)
and i need to
prove that it stays the same type by addition
and multiplication by scalar
and existence of zero.

field is like a formula
f(7)=a+7*b

is this correct

Read the wiki article I linked, what you wrote is not very mathematical in nature i.e. a field is definitely not "like a formula f(7) = a + 7*b". These are pretty advanced mathematical ideas and you should have at the very least the definition of each ingrained in your brain before you go out and try to prove things about them.